When to put new ball into play after HR?

[gsf23]

I also go after the batter-runner touches home. Take the ball out of my bag after he touches third, watch him/her touch home then give the ball to the catcher.

I also used to throw the ball back to the pitcher until this one game. First inning I see that the catcher is throwing one-hoppers back the the pitcher. I ask the catcher if he hurt his arm and he says no, the pitcher has a broken bone in his glove hand. So about top of the sixth, batter hits a ball out of play and, you guessed it, I pulled out a ball and fired a strike at the pitcher. he mananged to knock it down with the webbing of the glove and didn't hit his hand, but I quit throwing to pitchers after that.